There is no formula for when to use what movements in badminton. Your movements around the court to get to wherever the shuttle is going to be is supposedly to be as fluid as possible. So you can be as early as possible to play yr return shots properly.
You should try and split step just before the opponent hits the shuttle, so that you land as they hit it and then push off once you know where the shuttle is going. On serve you can't leave the ground (before they hit it) so bounce on your knees slightly as they hit the shuttle. The point of the split step is to pre load your muscles. If you get things rights you will notice you are moving a lot faster on court. Unfortunately it's one of those things that can take a bit of practise to get your timing right.
